Experimental mechanician for plate lattice metamaterial discovery

Abstract

Plate lattice metamaterials offer higher strength-to-weight efficiency than traditional truss lattices. Their optimization, however, is challenging when having more parameters and multiple objectives. Here, we present an experimental mechanician (ExMech) to identify lightweight and high-strength plate lattices. ExMech integrates a robotic platform for standardized data generation and a multi-objective active learning framework for guided exploration of structure–property relationships. It optimizes plate–truss hybrid lattices in a three-variable, three-objective space. It establishes the Pareto front in 25 iterations, reducing experimental workload significantly. Pareto-optimal combinations show 15.6% and 12.0% improvements in compressive and shear yield strength respectively without compromising lightweight metrics. When prioritizing strength, it yields 158.5% and 194.8% improvements with a 13.2% increase in weight. By analyzing variable-objective relationships and failure patterns, we could identify more structurally efficient combinations. The discovered lattices are demonstrated in compression–shear tunable 3D-printed midsoles. This study illustrates a self-optimizing experimentation approach for plate lattice discovery, accelerating the discovery of new mechanical metamaterials.

Here, the authors design a self-optimizing experimental system for the automatic discovery of lightweight and high‑strength plate‑lattice metamaterials, integrating a robotic platform with a machine learning framework.

Introduction

Inspired by natural architectures, artificial lightweight and high-strength materials have been extensively studied and applied^1–3. Lattice materials, as typical mechanical metamaterials, were initially focused on truss-based, open-cell structures^4,5 but have progressively shifted toward plate-based, closed-cell structures^6,7 and plate–truss hybrids^8–10. Benefiting from reduced configurational entropy through superior structural interconnectivity, plate lattices have demonstrated exceptional structural efficiency over truss lattices in strength-to-weight performance, even approaching the Hashin-Shtrikman and Suquet theoretical limits^11,12. However, plate lattice optimization faces a high-dimensional parameter challenge stemming from variable combinations and objective conflicts. Specifically, lattice variables span three levels: topological mode (e.g., arrangement pattern^13,14, hybridization type^9,15,16), geometric configuration (e.g., plate shape^17,18, hole layout^19,20), and dimensional parameter (e.g., cell size¹⁹, plate thickness^15,21), creating an extensive variable combination space. Also, objective conflicts across nonlinear properties (e.g., isotropy^7,22, strength^12,23, toughness^16,24, lightweight^22,25) complicate trade-off analyses, further intensifying this challenge. For instance, a 3-variable, 3-objective optimization sampled at 5% intervals requires at least 83,349 evaluations.

As a solution, computational simulations enable rapid and cost-effective large-scale dataset generation^20,26,27, yet reliability challenges persist due to micro-damages, material nonlinearity, and random defects^21,28,29. Alternatively, manual trial-and-error experimental approaches are equally limited by time-intensive exploration and batch-effect-induced credibility reductions, restricting experimental studies to small-scale³⁰, single-factor¹⁶, and confirmatory²¹ investigations that hinder the discovery of optimal lattice materials. Gratifyingly, the marriage between automation and artificial intelligence (AI) offers a new self-evolving solution, whose core concept combines “can-do” and “can-think” capabilities: the former produces experimental data through high-throughput and standardized protocols; the latter dynamically optimizes sequential experiments through an evolving understanding of design principles and objective trade-offs, drastically reducing the number of required experiments; this marriage enables human-like optimization of plate lattices across the aforementioned high-dimensional parameter space.

Recent materials science studies have demonstrated the above concepts, termed self-driving^31–34, AI-driven³⁵, autonomous laboratory^36,37, and robotic scientist^37–39. These have optimized diverse material properties including optical (e.g., chirality⁴⁰, photostability⁴¹), electrical (e.g., hole mobility³³, resistance⁴²), and mechanical properties (e.g., bond strength³⁴, toughness^43,44). For multi-objective optimization, some studies have examined trade-offs such as processing temperature versus conductivity in palladium films³², and activity versus stability in non-noble metal oxides⁴⁵, thus validating the feasibility of self-optimizing experimentation. However, in lattice materials, whether truss lattices or the newly developed plate lattices, no such implementations have been reported, let alone optimizing multiple conflicting objectives like low density versus high strength.

Here, we report an experimental mechanician, ExMech (Fig. 1), to discover lightweight, high-strength plate lattices, combining a robotic platform (9 workstations for standardized experimental data generation) and a multi-objective active learning framework (guiding directional exploration through evolving structure-property understanding). Our ExMech is developed to address the most complex plate lattices (plate–truss hybrid lattices^8–10 PTHL) for 3-objective optimization (compressive yield strength, shear yield strength, lightweight) in a 3-variable space (plate thickness, hole radius, strut diameter). Our ExMech establishes the Pareto front (PF) in just 25 iterations (150 mechanical tests), achieving a 370-fold reduction in experiment workload (compared to 55,566 mechanical tests). Without compromising lightweight metrics, Pareto-optimal combinations show 15.6% and 12.0% improvements in compressive and shear yield strength respectively. Alternatively, the mechanical performances can be prioritized (158.5% and 194.8% enhancement) with only 13.2% lightweight sacrifice. Furthermore, through analyzing variable-objective relationships and failure patterns, we identify structurally efficient combinations, revealing a new reinforcement mechanism termed strut-plate synergy. As an application, the discovered lattices are demonstrated in compression–shear tunable 3D-printed midsoles. We believe this study to be the very demonstration of a self-optimizing experimentation paradigm for plate lattice discovery, enabling accelerated development of mechanical metamaterials.

Fig. 1. Experimental mechanician ExMech.

a The ExMech composed of a robotic platform with 9 workstations and a multi-objective active learning framework. b Self-optimizing experimentation paradigm for plate lattice discovery, incorporating design-manufacture-characterize-learn cycles.

Results

Experimental mechanician

The experimental mechanician ExMech (Fig. 1a) integrated a robotic platform with 9 workstations and a multi-objective active learning framework, operating in concert to iteratively execute design-manufacture-characterize-learn cycles (Fig. 1b) without human intervention, constituting a self-optimizing experimentation paradigm for plate lattice discovery. Its workflow began with designing, where digital models were generated through parametric modeling and subsequently converted to printing files. In manufacturing, tasks were distributed to three 3D printing workstations. Once fabricated, specimens were automatically removed from the fabrication platform by the detaching module and sequentially transferred to the cleaning and drying workstations for post-processing. In characterizing, weighing, uniaxial compression testing, and short-beam shear testing were performed with dedicated workstations for objective acquisition. In learning, observed data were processed with soft constraints by a multi-objective active learning framework, and then used to train a Gaussian Process Regression (GPR) surrogate model. The GPR predictions informed an Expected Hypervolume Improvement (EHVI) acquisition function that recommends optimal combinations for the next designing, completing a workflow iteration. Herein, a transfer workstation (a 6-axis robot with electric gripper and depth camera) handled specimen positioning, grasping, and transferring tasks, interconnecting all workstations. Additionally, a software system, including application interfaces, algorithm frameworks, and workflow management, was primarily developed in Python. More information about the ExMech is provided in the section of Methods, Supplementary Figs. 1 and 2, and Supplementary Movie 1.

Establishment of the Pareto front

In the plate–truss hybrid lattices (PTHL) family, for the first time, the octet plate lattice (OCPL) was integrated with the simple cubic truss lattice (SCTL) under identical unit dimensions and spatial orientation (Fig. 2a). The face-centered nodes of the OCPL aligned with the strut terminals of the SCTL to enhance interconnectivity and fill the void body center, which could be envisioned to yield favorable mechanical-lightweight trade-offs. The ExMech manipulated three structural variables: plate thickness (PT), hole radius (HR), and strut diameter (SD). Notably, drainage holes (structural necessities for closed-cell additive manufacturing) simultaneously influenced both mechanical and lightweight properties through structural and processing effects by regulating resin evacuation efficiency. Each iteration included fabrication of three compression specimens (3×3×3 arrays) and three shear specimens (2×2×8 arrays) for triplicate measurements of quasi-static uniaxial compression testing and short-beam shear testing. Compressive yield strength σ_yc and shear yield strength σ_ys were automatically extracted from force-displacement curves via a custom analysis pipeline, while porosity η served as the lightweight metric. Experimental details are provided in the section of Methods.

Fig. 2. Establishment of the Pareto front (PF).

a Definition of variables (plate thickness, PT; hole radius, HR; and strut diameter, SD) and objectives (compressive yield strength, σ_yc; shear yield strength, σ_ys; and porosity, η), with the PTHL (plate–truss hybrid lattices) family composed of SCTL (simple cubic truss lattice) and OCPL (octet plate lattice). b 3D Pareto front and its 2D projection planes. c Ashby plots of the discovered PTHL combinations and their comparison with other lattice topologies, with data from experiments and FEA (finite element analysis). Source data are provided as a Source Data file.

Beginning with three random initial combinations, the ExMech operated autonomously for 60 hours, completing 55 iterations (25 iterations for convergence that will be analyzed in later sections) to ultimately generate a PF comprising 27 non-dominated solutions (Fig. 2b; error bar version in Supplementary Fig. 3). The PF demonstrated the trade-offs among three objectives: σ_yc, σ_ys, and porosity η. Representative structures from key PF points are displayed, revealing the complex variable-objective relationships. The 3D PF exhibited a descending gradient along the η axis and a roughly symmetric distribution in the σ_yc and σ_ys directions, suggesting pronounced trade-offs between σ_yc and η, as well as between σ_ys and η. In comparison, the trade-off between σ_yc and σ_ys was less significant and was strongly affected by η. This phenomenon is more clearly presented in three 2D projection planes. Moreover, when reducing dimensionality from 3D to 2D, the number of non-dominated solutions decreased to 14 or fewer, indicating significantly intensified trade-offs in higher-dimensional objective spaces.

Figure 2c presents the Ashby plot comparing relative compressive strength σ_yc/σ_m and relative shear strength σ_ys/σ_m versus relative density 1−η, which is commonly employed in the study of lattice materials for evaluating structural efficiency. Herein, σ_m represents the yield strength of the base material, with its specific value provided in Supplementary Table 1. Compared to the average of initial combinations, the non-dominated solutions obtained through the experiments demonstrated significant performance improvements. Specifically, without any sacrifice in lightweight metric, the combination from the 16th round exhibited a 15.6% increase in σ_yc and a 12.0% increase in σ_ys. The combination from the 21st round showed multiplied mechanical performance, achieving a 158.5% improvement in σ_yc and 194.8% improvement in σ_ys, with only a 13.2% loss in the lightweight metric. These results substantiate the possibility of enhancing one material property without significant compromise in another. Besides, to better demonstrate the improvement from the initial level, three initial combinations were compared separately across three objectives (Supplementary Table 2).

Moreover, the relationship between relative density and relative strength demonstrates the typical power-law behavior of lattice materials, implying that mechanical properties can be significantly enhanced by adjusting the relative density. Compared to other topological categories, the PTHL demonstrated superior compressive efficiency over truss^4,46 and shell architectures⁴⁷. The closed-cell octet plate²⁶, as predicted by finite element analysis (FEA), surpassed all subcategories including the PTHL, reaching the Suquet upper bound; however, its fabrication remains challenging in the real world without drainage holes^1,12. Honeycomb⁴⁸ also exhibited higher compressive efficiency at low densities, but their highly anisotropic mechanical responses (i.e., effectiveness limited to a single loading direction) severely restrict practical applications. In terms of shear yield strength σ_ys of plate lattices, experimental studies remain lacking. At low relative densities, the PTHL underperformed in shear efficiency compared to other topologies^17,49–52. However, as the relative density approached 0.41, the relative shear strength improved markedly, surpassing half of the topologies.

Multi-objective active learning framework

In addition to its precise identification of the PF, the ExMech demonstrated another remarkable capability in efficiency, achieving convergence to the PF boundary through a self-optimizing paradigm. As illustrated in Fig. 3a, the purpose of the ExMech is to navigate the high-dimensional variable space, approach the true PF of multiple objectives, and reveal relationships between variables and objectives. Herein, the high dimensionality originates from the combinations (i.e., 9,261) of variable types (i.e., PT, HR, SD) and their gradients (i.e., 0% to 100%); furthermore, the presence of multiple objectives (i.e., σ_yc, σ_ys, η) substantially increases the required mechanical tests (i.e., 55,566). Even high-throughput automated platforms struggle to adequately address such challenges. Accordingly, a multi-objective active learning framework (Fig. 3b) was developed to enable directional automated experiments, replacing random trial-and-error or brute-force search.

Fig. 3. Multi-objective active learning framework.

a Purpose of the framework to provide a navigation for a high-dimensional variable space with multiple objectives. b Illustration of the framework workflow, using GPR (Gaussian Process Regression) as the surrogate model and EHVI (Expected Hypervolume Improvement) as the acquisition function.

In Stage I, observations of candidates acquired from the last iteration were incorporated into the labeled dataset, followed by normalization and constraint processing of the objective values. In Stage II, a new training dataset was generated with objective functions: O₁, O₂, and O₃. Specific definitions of objective functions and constraints are provided in the section of Methods. In Stage III, the surrogate model was trained using the dataset, where the distribution of each objective function was modeled with an independent GPR^32,53. In Stage IV, the PF based on the current dataset was constructed, while the trained surrogate model was applied to generate predictions of objectives for a batch of candidates. In Stage V, values of the acquisition function for each candidate were computed using the above predictions. The EHVI (i.e., the anticipated PF expansion) was adopted here, which is a widely used acquisition function in multi-objective optimization engineering^54,55. Stages Ⅳ and Ⅴ were repeatedly executed, starting a local optimization procedure from various initial combinations to provide a series of local maxima. The best candidate among all local optimization results was selected for experimental observation. In Stage Ⅵ, objectives of the recommended combination were obtained by executing the design-manufacture-characterize procedure. The above framework operated in a closed loop, using hypervolume⁵⁶ (representing the volume covered from a reference point to the PF) to evaluate the solution quality, thereby approaching the true PF through maximizing the hypervolume. Detailed interpretation of the framework (e.g., description of GPR and EHVI), as well as additional information (e.g., reference point, kernel functions, and hyperparameters), is presented in the section of Methods and Supplementary Note 1.

Beyond direct constraints on variables, soft constraints were also applied to objectives due to requirements from real-world engineering^57–59. For the ExMech, this soft constraint strategy prevented issues from extreme conditions (e.g., physical infeasibility, exceeding equipment limits), thereby guiding the framework away from infeasible regions and avoiding wasted time caused by repeated exploration. Notably, the soft constraint strategy penalized combinations that violated constraints rather than discarding them, as validated and discussed in detail in the next section. Accordingly, the essence of the framework lies in an iterative process where the surrogate model simultaneously learns objective functions and constraint relationships.

Self-optimizing mechanisms

This section presents the concrete process of plate lattice discovery without human intervention, as well as the mechanism discussion. Fig. 4a illustrates the evolution of hypervolume over the iteration process (red curve), with min-max normalization applied for ease of comparison. Starting from three random combinations, the PF converged within only 25 iterations (150 mechanical tests), achieving a 370-fold reduction in experiment workload compared to the original 55,566 mechanical tests. The hypervolume curve can be roughly divided into four stages, with corresponding PF and sampling displayed in the subplots. From Stage Ⅰ to Ⅲ, the boundary region was rapidly established with a small number of exploratory trials, corresponding to the steep rise in the hypervolume curve. In Stage Ⅳ, exploration concentrated along the existing boundary contours, contributing little to its expansion and resulting in a plateau in the hypervolume curve (55 iterations in total). Notably, although 14 non-dominated solutions were identified in the last 30 iterations, the hypervolume increased only marginally by 0.33%, indicating that a relatively complete PF can be obtained within about 25 iterations. More interestingly, the PF expanded preferentially along the porosity direction due to a pronounced competition between porosity and strength. Once approaching the boundary in this direction, the hypervolume exhibited diminishing returns, leading to a lateral expansion of the PF, i.e., the trade-off between σ_yc and σ_ys. Raw data, including structural variables, objective values, and non-dominated solutions, are provided in Supplementary Table 3. In addition, to examine the influence of initial values, a reference case was conducted with different initial combinations (gray curve; its PF is shown in Supplementary Fig. 4, and the raw data are provided in Supplementary Table 4). The results show faster convergence occurring at the 18th iteration, along with an overlapping final hypervolume value, indicating the role of initial values in convergence speed. Further, to examine the influence of initialization sizes on convergence, multiple in silico virtual experiments were performed (see Supplementary Note 2 and Supplementary Figs. 5 and 6 for details).

Fig. 4. Self-optimizing mechanism.

a Evolution of hypervolume over iterations in objective space. b Evolution of sampling activities over iterations in variable space. c Cross-sections of response surfaces reflecting variable-objective relationships. Source data are provided as a Source Data file.

Beyond focusing on iteration patterns in objective space, it is more critical to understand how the ExMech learns variable-objective relationships and handles constraints. Figure 4b illustrates the location of the sampled combinations in the variable space, where colors indicate iteration sequences. The grey region represents the constraint space, where the box planes (HR = 0.2, SD = 2.0 and 0.01, PT = 0.05 and 0.6) represent the input constraints, and the left surface embodies the output constraint in the variable space. Unlike the predefined constraints of variables, the surface of objective constraints is complex and implicit, requiring thorough exploration and learning of constraint patterns for delineation. It can be observed that in the HR direction, early-stage (blue points) and mid-stage (grey points) iterations conducted limited sampling within the constraint region before shifting outside, with scarce entry into the constraint region during later stages. This phenomenon demonstrates that constraints effectively guided the exploration path, narrowing the space awaiting exploration, which accounts for 52.05% of the entire space. Additionally, the top view and side view of the variable space are presented in Supplementary Fig. 7 to demonstrate sampling activities in the other PT and SD directions.

Moreover, response surfaces, derived by regressing each GPR model against experimental outputs, provide intuitive representations of input-output relationships. Since the response surfaces of a 3D input space form a 4D hypersurface, cross-sections (i.e., H1, H2, and H3 in Fig. 4b) of the response surfaces were depicted for visualization in Fig. 4c. Each row represents the response surfaces of an independent objective function, with color intensity corresponding to the function value. The peaks for σ_yc and σ_ys lay in the upper-right corner, while the peak for porosity lay in the lower-left corner, showing that PT and SD are positively correlated with mechanical strength but negatively correlated with porosity. However, HR exhibited the opposite trend, as graphs of σ_yc and σ_ys showed a global fading in color with HR increasing from 0.8 mm to 1.2 mm, while this trend was entirely reversed for η. Furthermore, as HR increased, multiple local optima emerged, and the contour orientation shifted from horizontal to vertical. These findings suggest that the variable-objective relationships are intricate, characterized by strong nonlinearity, likely arising from coexisting synergistic-antagonistic coupling effects. Additional cross-sections of response surfaces in the other PT and SD directions are presented in Supplementary Fig. 8.

Further, for comprehensive comparison with purely computational optimization, we developed a simulation-only pipeline (i.e., FEA-guided design) initialized from identical starting points as the experiment approach. Simulation configuration, results, and analysis, are detailed in Supplementary Methods, Supplementary Table 5, and Supplementary Note 3. Results confirm simulation-guided optimization’s applicability to lattice design. While exhibiting faster hypervolume convergence than experiment-guided optimization, the simulation approach shows notable differences in final convergence values. The accelerated convergence stems from eliminated noise interference and simplified nonlinear factors (e.g., process-dependent effects, damage evolution, fracture behavior). Convergence value differences arise from simulation-experiment deviations and constraint relaxation (i.e., expanded feasible region). These findings suggest a hybrid strategy: first leveraging simulation’s rapid convergence, then refining results through experimental optimization.

Variable-objective interaction mechanisms

Although the response surfaces offer an overview of variable-objective relationships, the key driving factors remain unclear. Here, SHapley Additive exPlanations (SHAP), a quantitative method to decipher the black-box nature of AI machine learning, was employed to quantify variable importance for the objectives over iterations (Fig. 5). The left waterfall plots illustrate the variation in the contribution rates of variables (HR, PT, SD) to objectives (σ_yc, σ_ys, η) over iterations, where the global contribution was calculated as the mean absolute Shapley value of each variable across all combinations. For each objective, fluctuations in contribution rates of the three variables gradually decreased from the 25th iteration and remained stable after the 30th iteration, suggesting the ExMech successfully learned the underlying mechanisms of feature influence objectives. For σ_yc, contributions from the three features were comparatively balanced. For σ_ys, PT exerted the most significant influence, while for η, HR dominated with a contribution exceeding 50%.

Fig. 5. Variable-objective interaction mechanisms.

SHAP (SHapley Additive exPlanations) analysis for quantifying variable-objective relationships over iterations (left), with the distribution of Shapley values displayed in violin plots (right). Source data are provided as a Source Data file.

The final dataset from the left waterfall plots is further used to generate the right violin plots, illustrating the distribution of all combinations in terms of Shapley values. Here, variables are ranked by influence, with the horizontal axis indicating both the magnitude and direction of variable effects, while the vertical width illustrates the clustering density of combinations. The color distribution reveals that PT and SD were both directly correlated with σ_yc and σ_ys, while HR was inversely correlated with σ_yc and σ_ys, with this trend reversed for η. A wide horizontal spread and noticeable color crossover are observed in the plots of σ_yc and σ_ys, particularly for HR and PT, revealing the existence of interaction effects among variables, as similar variable values yielded vastly differing contributions to objectives, even opposite effects. Moreover, an obvious asymmetric distribution existed, especially for the graph of η, where η responded only to higher values of HR, indicating a strong nonlinear influence of HR on η. This is because HR plays a dual role: a design-dependent impact on geometric porosity and a process-dependent effect on resin evacuation.

Mechanical mechanisms

To further investigate material failure behaviors and thereby uncover underlying mechanical mechanisms, the load-displacement curves from compression and shear tests are presented in Fig. 6a. For both compression and shear tests, yield strength generally showed a positive correlation with density. Notably, failure strain aligned with density changes in compression tests, i.e., the yield point appeared later as density increased. Conversely, in shear tests, low-density specimens showed delayed yielding or even no distinct yield point, as their reduced stiffness promoted large deformations rather than failure.

Fig. 6. Mechanical mechanisms.

a Load-displacement curves in compression and shear tests. b Two representative combinations with distinct relative densities in compression tests. c Two representative combinations with distinct relative densities in shear tests. d Strut-plate synergistic reinforcement mechanism. Source data are provided as a Source Data file.

Two representative combinations with distinct relative densities (Combinations 48 and 17 in Supplementary Table 3) are selected, and their complete load-displacement curves along with deformation images from compression tests are displayed in Fig. 6b (more images are shown in Supplementary Fig. 9). The deformation exhibited sequential stages typical of lattice structures: elastic deformation, plastic deformation, and densification³⁰, with failure initiated by localized buckling occurring at the bottom layer. For low-density Combination 48, failure occurred at an earlier stage (13.75% strain) due to buckling-induced instability (orange lines). As buckling deformation propagated upward, sequential buckling and folding occurred layer by layer (30.5% strain), accompanied by localized fractures, corresponding to the gradual descending segment of the load-displacement curve. For high-density Combination 17, it remained in the elastic deformation stage at 13.75% strain, with failure occurring later. As strain increased, intensified buckling deformation caused significant stress concentration, leading to localized fractures (yellow area). The failure expanded rapidly and propagated from the bottom to top layers, resulting in simultaneous fracture failures at multiple locations (Supplementary Movie 2), manifested as a second ascending linear segment on the load-displacement curve. This is primarily due to the substantial densification presented before failure, resulting in rapid stress redistribution after localized failure occurred, thus manifesting macroscopically as maintained load-bearing capacity. Additionally, localized stress concentrations around fractured regions induced directional crack propagation, forming a macroscopic shear band (i.e., a stress concentration zone as illustrated by white dashed lines at 30.5% strain). The localized densification within this stress concentration zone introduced a strengthening effect that mitigated collapse behavior impacts.

Also, the load-displacement curves and deformation images of the two representative combinations from shear tests are displayed in Fig. 6c. The low-density Combination 48 exhibited an elastic-plastic shear response. Its load-displacement curve slightly declined at 25.0% deformation but avoided fracture until the end (64.5% deformation), as energy was absorbed through large deformation. For high-density Combination 17, progressive deformation induced tangential plastic flow (25.0% deformation) and further evolved into fractures along the shear direction, with cracks rapidly propagating across the entire cross-section (54.0% deformation), ultimately causing complete fracture and separation. Correspondingly, the load-displacement curve rapidly dropped after reaching its peak, reflecting a mixed mechanism of yielding and brittle fracture.

The above two combinations demonstrate a mixed failure mode of structural instability and brittle fracture. The low-density combination is dominated by buckling instability with energy absorbed via large deformation, showing elastic-plastic behaviors; while the high-density combination is dominated by brittle fracture with energy dissipated through fracture, exhibiting brittle behaviors. This suggests that the yielding behaviors of PTHL can be modulated by adjusting the density, while being challenged by the question that how to achieve minimal strength loss when reducing density. Here, specific strength is introduced to characterize the structural efficiency of the PTHL, defined as the ratio of strength to density. Through comparative analysis of structural features and specific strengths (detailed discussion provided in Supplementary Note 4), it was found that combinations with higher specific strengths tended to maintain a moderate ratio between OCPL and SCTL. Even at similar density levels, a significant reduction in structural efficiency occurs when either the plate elements or strut elements become too weak, indicating a synergistic effect between these two components.

Specifically, the strut-plate synergy is illustrated in Fig. 6d. The configuration of the unit cell has improved interconnectivity, with the SCTL connecting the face centers (white dashed circles) and filling the hollow body center, enabling efficient load transfer between adjacent unit cells. This configuration helps restrain OCPL’s transverse expansion arising from the Poisson effect (blue arrows), thereby reducing plate bending and mitigating structural collapse. Besides, SCTL’s loading efficiency was improved in this configuration: 1/3 of the struts sustain compressive stress to resist axial deformation and 2/3 bear tensile stress to constrain transverse deformation (orange arrows). In contrast, in SCTL alone, only 1/3 of the struts are engaged in load-bearing. Therefore, the strut-plate synergy explains the observed detachment at the strut-plate junction during the critical failure stage (Combination 17 under 25% strain, see Supplementary Movie 2), due to severe stress concentration at the junctions. Additionally, combinations violating the strut-plate synergy typically exhibit diminished structural efficiency. For example, oversized holes or excessively thin plates result in degeneration into truss; while overly slender struts degrade PTHL to plate lattices.

Midsole applications

The PF of the PTHL family, mapped through self-optimizing discovery, delineates clear trade-off boundaries for material design, thereby facilitating decision-making in choosing materials best suited for specific applications. 3D-printed footwear features advantages including customization (size, stiffness, and strength), recyclability, and low cost, and has been widely employed in orthopedic shoes and performance props^60,61. The midsole, as the primary load-bearing structure, presents the most challenging design requirements since it continuously withstands dynamic impacts and multidirectional loads, demanding stringent performance in compressive strength, shear strength, and lightweight properties. Here, leveraging the comprehensive understanding of the PTHL family, appropriate members can be selectively employed as fundamental unit cells for midsole design. For demonstration purposes, two combinations on the PF with a porosity of 76.5% ± 0.5% (higher than mainstream commercial midsole materials) were selected (Fig. 7a): Combination 0 favoring shear resistance and Combination 32 favoring compression resistance. These were fabricated via 3D printing (Fig. 7b). The design and fabricating methods are detailed in the section of Methods and Supplementary Fig. 10.

Fig. 7. Midsole applications.

a Two combinations selected from the established PF for the cell design of midsoles. b Photographs of the midsoles fabricated via 3D printing. c Failure patterns of the midsoles under normal-shear loads.

Tests were conducted on both midsoles at an angle of 35° to simultaneously apply normal and shear loads (Fig. 7c). The damaged regions of both midsoles were primarily concentrated around the forefoot metatarsal area, which experienced high plantar pressure. Combination 0 exhibited compressive-dominated deformation, characterized by buckling collapse in the normal direction. This resulted in a nearly symmetrical failure pattern, marked by fractures in plates surrounding the face centers of lattice cells (red forks). In contrast, Combination 32 displayed shear-driven deformation, manifested as interlayer sliding and misalignment. A zigzag shape (yellow line) formed at the contact interface due to deflection from tangential loads, producing a distinct asymmetrical failure pattern as fractured plates in each lattice aligned on the same side.

Discussion

In summary, we developed an experimental mechanician ExMech to discover lightweight, high-strength plate lattices, combining a robotic platform (9 workstations for standardized experimental data generation) and a multi-objective active learning framework (guiding directional exploration through evolving structure-property understanding). Our ExMech was tasked with optimizing plate–truss hybrid lattices for 3-objective optimization (compressive yield strength, shear yield strength, lightweight) in a 3-variable space (plate thickness, hole radius, strut diameter). The ExMech established the Pareto front in just 25 iterations (150 mechanical tests), achieving a 370-fold reduction in experiment workload (compared to 55,566 mechanical tests). Without compromising lightweight metrics, Pareto-optimal combinations showed 15.6% and 12.0% improvements in compressive and shear yield strength respectively. Alternatively, the mechanical performances can be prioritized (158.5% and 194.8% enhancement) with only 13.2% lightweight sacrifice. Furthermore, through analyzing variable-objective relationships and failure patterns, we identified structurally efficient combinations and revealed a new reinforcement mechanism termed strut-plate synergy. As an application, the discovered lattices were demonstrated in compression–shear tunable 3D-printed midsoles. We believe this study to be the very demonstration of a self-optimizing experimentation paradigm for plate lattice discovery, with the potential to expand to more complex metamaterials (e.g., topological and hierarchical lattices), enabling accelerated development of mechanical metamaterials.

Methods

Experimental mechanician

The ExMech was designed and assembled in a dedicated area within our laboratory, consisting of 9 key workstations, 2 computers, and a multi-objective active learning framework to perform a closed-loop design-manufacture-characterize-learn cycle without human intervention. The workstations were composed of self-designed devices/modules and commercial equipment, equipped with self-developed application interfaces following a unified template. Specific organization, device information, and communication architecture of the ExMech are provided in Supplementary Fig. 1. Besides hardware, a software system was developed using Python, responsible for device interaction, task management, data storage, and analysis, with the flowchart detailed in Supplementary Fig. 2.

Specimen design

The parametric modeling approach was employed to automatically generate STL (Standard Triangle Language) files of the PTHL, according to the variable combinations (PT, HR, and SD) designed by the multi-objective active learning framework. Specifically, the parametric modeling script was developed with Python and 3D modeler to create solid objects of compression and shear specimens separately, which were then converted into STL files and stored in the task queue.

Specimen manufacture

The main program first queried the status of the printer cluster. Once idle 3D printing workstations are detected, executable printing files (the.form files) of the latest task are automatically generated and assigned. Then, specimens were fabricated from photosensitive resin ClearV4 (Formlabs Inc., USA) using a Stereolithography printer (Form 3 + , Formlabs Inc., USA) with a layer thickness of 50 µm and an optical wavelength of 405 nm. For each combination, three compression specimens (3×3×3 array) and three shear specimens (2×2×8 array) will be fabricated, with a cell size of 8 mm × 8 mm × 8 mm and a minimum feature size of 50 µm. Once fabricated, specimens are transferred by the detaching module into a custom-designed loading basket for convenient transportation and positioning, with dimensions of 80 mm × 80 mm × 100 mm.

Specimen post-processing

Post-processing is essential to remove residual resin adhering to the surface and internal cavities of the specimens, in case of interference with subsequent characterizing. Specimens were first transferred by a 6-axis robot to the cleaning workstation, followed by an ultrasonic cleaning (SN-J10, ShenNa, China) at 40 kHz for 15 min with isopropanol (Macklin Inc., China) as the cleaning solution. Subsequently, the specimens, along with the loading basket, are placed in the drying workstation to remove residual isopropanol. The drying task is executed by an air-blow module (AIKER, China) with an airspeed of 82 m/s, lasting for 2.5 minutes.

Specimen characterizing

Before characterizing, the specimens were first poured from the basket and dispersed, then individually picked up through a vision-guided procedure. Porosity η was first measured using the weighting workstation, calculated by the formula:

$$\eta =\left\{\begin{array}{c}1-\frac{m}{27V{\rho }_{\mathrm{m}}}\mathrm{compressionspecimen}\hfill \\ 1-\frac{m}{32V{\rho }_{\mathrm{m}}}\mathrm{shearspecimen}\hfill \end{array}\right),$$ 1

where V = 512 mm³ represents the volume of a single cell and ρ_m denotes the density of the resin. The Quasi-static uniaxial compression test was conducted by a universal testing machine (Tian Yuan Test instrument, China) in the compression test workstation, with a capacity of 5 kN and a resolution of 0.03%. The test procedure was set with a constant speed of 8 mm/min and termination conditions of 60% strain or exceeding the load capacity. Once the test is finished, the specimen is removed from the test platen by a sweeping module. The compressive yield strength σ_yc was calculated by:

$${\sigma }_{\mathrm{yc}}=\frac{F_{\mathrm{c}}}{l^2},$$ 2

where F_c represents the failure load, and l = 24 mm denotes the length of compression specimen. The shear test was performed following the short-beam shear test procedure^62–64, using another universal testing machine (ZHIQU Precision Instruments, China), with a capacity of 2 kN and a resolution of 0.01%. A three-point bend fixture was installed, with the roller radius of 5 mm, span height of 50 mm, and span of 32 mm (to ensure a span-to-depth ratio of 2). The loading member moved downward at a constant speed of 6 mm/min until a load drop-off exceeding 30% occurs or the head travel exceeds the specified value of 10.5 mm. The shear yield strength σ_ys can be calculated as:

$${\sigma }_{\mathrm{ys}}=\frac{3}{4}\frac{F_{\mathrm{s}}}{bh},$$ 3

where F_s denotes the failure or maximum load. b and h represent the width and thickness of the test specimen, respectively, and b = h = 16 mm in this study. All mechanical tests were repeated three times to obtain reproducible results. Note that F_c and F_s were automatically extracted from the force-displacement curves using a data analysis pipeline customized with Python. Raw data from the characterizing stage was saved in a local database on computer 2, while the processed objectives (e.g., σ_yc, σ_ys, and η) were sent to computer 1, thereby updating the labeled database for model training.

Multi-objective active learning framework

Two critical components of multi-objective active learning framework are the surrogate model and the acquisition function. For the surrogate model, each objective function was modeled independently by a GPR model, choosing a Gaussian kernel as the kernel function⁶⁵, with the length scale automatically optimized during each iteration. For the acquisition function, the reference point was set to [0, 0, 0], and the objective data were normalized to the range [0, 1]. The acquisition function was implemented through multiple restarts of a local optimization algorithm from diverse initial points, where the number of initial points was set to 10, with 5 restarts and 30 candidates per batch. These hyperparameters were determined based on the throughput and computational limitations, aiming to minimize the idle time of experimental equipment caused by prolonged runtimes of optimization algorithm. More details of the multi-objective active learning framework are supported in Supplementary Note 1.

Objective functions and constraints

The objective functions and constraints can be described as:

$$\max \left\{\begin{array}{c}{O}_1=\frac{{\sigma }_{\mathrm{yc}}}{{\sigma }_{\mathrm{m}}}+p\left({\sigma }_{\mathrm{yc}},{\sigma }_{\mathrm{ys}},\rho \right)\hfill \\ O_2=\frac{{\sigma }_{\mathrm{ys}}}{{\sigma }_{\mathrm{m}}}+p\left({\sigma }_{\mathrm{yc}},{\sigma }_{\mathrm{ys}},\rho \right)\hfill \\ O_3=1-\frac{\rho }{{\rho }_{\mathrm{m}}}+p\left({\sigma }_{\mathrm{yc}},{\sigma }_{\mathrm{ys}},\rho \right)\hfill \end{array}\right),$$ 4

$$\mathrm{st}\left\{\begin{array}{c}0\mathrm{.2mm}<=\mathrm{HR}<=\mathrm{1}\mathrm{.3mm}\hfill \\ 0\mathrm{.05mm}<=\mathrm{PT}<=\mathrm{0}\mathrm{.7mm}\hfill \\ 0\mathrm{.1mm}<=\mathrm{SD}<=\mathrm{2mm}\hfill \end{array}\right)$$ 5

Here, the goal of optimization is to maximize objective functions O₁, O₂, and O₃, subject to box constraints on 3 structural variables (PT, HR, and SD) and soft constraints on 3 objectives. The penalty function p is incorporated to penalize constraint violations, formulated as:

$$p\left({\sigma }_{\mathrm{yc}},{\sigma }_{\mathrm{ys}},\rho \right)=k{I}_{\mathrm{\rho }}\frac{\rho }{{\rho }_{\mathrm{m}}}+k{I}_{\mathrm{c}}{\sigma }_{\mathrm{yc}}+k{I}_{\mathrm{s}}{\sigma }_{\mathrm{ys}},$$ 6

where k is the penalty coefficient. ρ_m and σ_m are the intrinsic density and yield strength of the base material, respectively. The indicator functions I_ρ, I_c, and I_s are defined as follows:

$$I_{\mathrm{\rho }}=\left\{\begin{array}{cc}0,\hfill & \rho \le {\rho }_0\hfill \\ 1,\hfill & \rho >{\rho }_0\hfill \end{array}\right),I_{\mathrm{c}}=\left\{\begin{array}{cc}0,\hfill & {\sigma }_{\mathrm{yc}}\ge {\sigma }_{\mathrm{c0}}\hfill \\ 1,\hfill & {\sigma }_{\mathrm{yc}}<{\sigma }_{\mathrm{c0}}\hfill \end{array}\right),I_{\mathrm{s}}=\left\{\begin{array}{cc}0,\hfill & {\sigma }_{\mathrm{ys}}\ge {\sigma }_{\mathrm{s0}}\hfill \\ 1,\hfill & {\sigma }_{\mathrm{ys}}<{\sigma }_{\mathrm{s0}}\hfill \end{array}\right)$$ 7

The ρ₀, σ_c0, and σ_s0 denote the constraints on relative density, compressive, and shear strength, respectively, with all the specific values listed in Supplementary Table 1.

Testing procedure of the midsole

The midsole was placed on a platform inclined at 35° to simultaneously apply normal and shear loads. To achieve a destructive testing condition, the participant, weighing 70 kg, performed a single-leg stance. The test began with the participant placing the entire right sole in contact with the midsole. Subsequently, the heel was slowly lifted, reducing the contact area until only the forefoot maintained contact. Finally, the load was rapidly removed, and the damaged regions were recorded and photographed after testing. The full testing procedure is documented in Supplementary Movie 3. Note that the mechanical testing performed on the midsole is qualitative and destructive, evaluated through the observation of failure patterns. For further quantitative assessment of other mechanical properties (e.g., energy absorption, fatigue resistance), specialized midsole testing equipment and standards could be considered.

Author contributions

S.H.: Conceptualization, Funding acquisition, Supervision, Writing - review and editing. H. L.: Data curation, Investigation, Methodology, Software, Resources, Validation, Visualization, Writing—original draft. W.L.: Methodology. T.X.: Methodology. X.D.: Methodology. X.Z.: Methodology. X.S.: Supervision. Z.P.: Funding acquisition, Supervision. X.C.: Supervision.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The raw data generated in this study have been deposited in the GitHub database (https://github.com/haoran-sjtu/ExMech.git), without any restriction. Source data are provided with this paper.

Code availability

All custom code used in this study was based on open-source Python packages, which have been deposited in the repository of GitHub (https://github.com/haoran-sjtu/ExMech.git).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-70675-x.